aerodynamic models for darrieus-type straight-bladed vertical axis wind turbines

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Renewable and Sustainable Energy Reviews

12 (2008) 1087–1109

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Aerodynamic models for Darrieus-typestraight-bladed vertical axis wind turbines

Mazharul Islam�, David S.-K. Ting, Amir Fartaj

Department of Mechanical, Automotive and Materials Engineering, University of Windsor,

Windsor, Ont., Canada N9B 3P4

Received 20 October 2006; accepted 31 October 2006

Abstract

Since ancient past humans have attempted to harness the wind energy through diversified means

and vertical axis wind turbines (VAWTs) were one of the major equipment to achieve that. In this

modern time, there is resurgence of interests regarding VAWTs as numerous universities and

research institutions have carried out extensive research activities and developed numerous designs

based on several aerodynamic computational models. These models are crucial for deducing

optimum design parameters and also for predicting the performance before fabricating the VAWT.

In this review, the authors have attempted to compile the main aerodynamic models that have been

used for performance prediction and design of straight-bladed Darrieus-type VAWT. It has been

found out that at present the most widely used models are the double-multiple streamtube model,

Vortex model and the Cascade model. Each of these three models has its strengths and weaknesses

which are discussed in this paper.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Renewable energy; Vertical axis wind turbine; VAWT; Wind; Straight-bladed; Darrieus

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088

2. Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090

3. Modern VAWT types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091

see front matter r 2006 Elsevier Ltd. All rights reserved.

.rser.2006.10.023

nding author. Tel.: +1519 253 3000x2635; fax: +1 519 973 7007.

dress: [email protected] (M. Islam).

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ARTICLE IN PRESSM. Islam et al. / Renewable and Sustainable Energy Reviews 12 (2008) 1087–11091088

3.1. Savonius wind turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091

3.2. Darrieus wind turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092

3.3. H-Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094

4. General mathematical expressions for aerodynamic analysis of straight-bladed Darrieus-type

VAWTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094

4.1. Variation of local angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094

4.2. Variation of local relative flow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096

4.3. Variation of tangential and normal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097

4.4. Calculation of total torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097

4.5. Power output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098

5. Computational models for Darrieus-type straight-bladed VAWT . . . . . . . . . . . . . . . . 1098

5.1. Momentum model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098

5.1.1. Single streamtube model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098

5.1.2. Multiple streamtube model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100

5.1.3. Double-multiple streamtube model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102

5.2. Vortex model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103

5.3. Cascade model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108

1. Introduction

At present, there are two categories of modern wind turbines, namely horizontal axiswind turbines (HAWTs) and vertical axis wind turbines (VAWTs), which are used mainlyfor electricity generation and pumping water. The main advantage of VAWT is its singlemoving part (the rotor) where no yaw mechanisms are required, thus simplifying the designconfigurations significantly. Blades of straight-bladed VAWT may be of uniform sectionand untwisted, making them relatively easy to fabricate or extrude, unlike the blades ofHAWT, which should be twisted and tapered for optimum performance. Furthermore,almost all of the components of VAWT requiring maintenance are located at the groundlevel, facilitating the maintenance work appreciably.From the past experiences, it is evident that wind turbines can compete with

conventional sources in niche markets, and lower costs make them affordable options inincreasingly large markets. Environmentally benign VAWTs can be utilized for a range ofapplications, including (i) electricity generation; (ii) pumping water; (iii) purifying and/ordesalinating water by reverse osmosis; (iv) heating and cooling using vapour compressionheat pumps; (v) mixing and aerating water bodies; and (vi) heating water by fluidturbulence. In general, VAWT can sensibly be used in any area with sufficient wind, eitheras a stand-alone system to supply individual households with electricity and heat, or forthe operation of freestanding technical installations. If a network connection is available,the energy can be fed in, thereby contributing to a reduction in electricity costs. In order tomaximize the security of the energy supply, different types of VAWT can be supplementedby a photovoltaic system or a diesel generator in a quick and uncomplicated fashion.Through the combination of several VAWTs with other renewable energy sources and abackup system, local electrical networks can be created for the energy supply of smallsettlements and remote locations.

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Abbreviations and Acronyms

A projected frontal area of turbineAR aspect ratio ¼ H/CC blade chordCd blade drag coefficientCdor reference zero-lift-drag coefficientCD turbine overall drag coefficient ¼ FD=rAV 2

1

CDD rotor drag coefficient ¼ FD=rAV 21

Cl blade lift coefficientCn normal force coefficientCP turbine overall power coefficient ¼ Po=rAV 3

1

CQ turbine overall torque coefficient ¼ Q=rAV 21R

Ct tangential force coefficientd minimum distance from the vortex filamentD blade drag force~e unit vectorF force on blade airfoilFD turbine overall drag forceFn normal force (in radial direction)Ft tangential forceFta average tangential forceFt non-dimensional tangential force ¼ Ct (W/VN)2

H height of turbineHAWT horizontal axis wind turbineki exponent in the induced velocity relationL blade lift force_m mass flow rate

N number of bladep static pressurePo overall powerPN atmospheric pressureQ overall torque~r unit vectorR turbine radiusRe local Reynolds number ¼WC/ut blade spacing ¼ (2pR/N)V centre line velocity along freestream velocity directionVa induced velocityVad induced velocity in the downstream sideVau induced velocity in the upstream sideVc chordal velocity componentVcd chordal velocity component in the downstream sideVcu chordal velocity component in the upstream sideVe wake velocity in upstream sideVn normal velocity componentVnd normal velocity component in the downstream side

M. Islam et al. / Renewable and Sustainable Energy Reviews 12 (2008) 1087–1109 1089

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Vnu normal velocity component in the upstream side~Vp induced velocity at a point P on the filamentVw wake velocity in downstream sideVG velocity contributed by circulationVN wind velocityVAWT vertical axis wind turbineW relative flow velocityWd relative flow velocity in the downstream sideWu relative flow velocity in the upstream sidea blade angle of attackad blade angle of attack in the downstream sideau blade angle of attack in the upstream sideg blade pitch angleG circulation per unit lengthy azimuth anglel tip speed ratio ¼ Ro/VN

n kinematic viscosityr fluid densitys solidity ¼ NC/Ro angular velocity of turbine in rad/s

M. Islam et al. / Renewable and Sustainable Energy Reviews 12 (2008) 1087–11091090

In this modern time, there is resurgence of interests regarding VAWTs as severaluniversities and research institutions have carried out extensive research activities anddeveloped numerous designs based on several aerodynamic computational models. Thesemodels are crucial for optimum design parameters and also for predicting the performancebefore fabricating the models or prototypes. In this paper, the authors attempt to explorethe main aerodynamic models that have been used for performance and design analysis ofstraight-bladed Darrieus-type VAWT through literature survey that are organized andbriefly described in the subsequent headings.

2. Historical background

Wind energy systems have been used for centuries as a source of energy for mankind.According to historic sources, the Babylonian emperor Hammurabi used windmills for anambitious irrigation project as early as the 17th century BCE [1]. Later on, Persianinventors developed a wind-power machine, a more advanced windmill than thatdeveloped by the Babylonians [2]. Arab geographers traveling in Afghanistan in the 7thcentury wrote descriptions of windmills, which resembled our modern revolving doors [3].Vertical windmills of this category were still used in some parts of Iran and Afghanistan inthe late 1980s, and it was estimated that they generated about 75 hp and can grind a ton ofwheat every 24 h [4].The earliest European windmills appeared in France and England in the 12th century

and quickly spread throughout Europe. These early wood structures, called post mills,were rotated by hand around a central post to bring the sails into the wind. By the late partof the 13th century the typical ‘European windmill’ had been developed and this becamethe norm until further developments were introduced during the 18th century. At the end

ARTICLE IN PRESSM. Islam et al. / Renewable and Sustainable Energy Reviews 12 (2008) 1087–1109 1091

of the 19th century there were more than 30,000 windmills in Europe, used primarily forthe milling of grain and water pumping [1].

3. Modern VAWT types

There have been many designs of vertical axis windmills over the centuries and currentlythe vertical axis wind turbines can be broadly divided into three basic types, namely (1)Savonius type, (2) Darrieus type, and (3) H-Rotor type. Brief descriptions of these VAWTtypes are given below.

3.1. Savonius wind turbine

The Savonius-type VAWT, as shown in Fig. 1, was invented by a Finnish engineerS.J. Savonius in 1929 [5]. It is basically a drag force driven wind turbine with two cups orhalf drums fixed to a central shaft in opposing directions. Each cup/drum catches the windand so turns the shaft, bringing the opposing cup/drum into the flow of the wind. This cup/drum then repeats the process, causing the shaft to rotate further, thus completing a fullrotation. This process continues all the time the wind blows and the turning of the shaft isused to drive a pump or a small generator. Though typical values of maximum powercoefficient for other types of wind turbines vary between 30% to 45%, those for theSavonius turbines are typically not more than 25% according to most investigators [6].This type of turbine is suitable for low-power applications and they are commonly used forwind speed instruments.

Fig. 1. Savonius-type VAWT.


3.2. Darrieus wind turbines

The modern Darrieus VAWT was invented by a French engineer George Jeans MaryDarrieus. He submitted his patent in 1931 [7] in the USA which included both the‘‘Eggbeater (or Curved Bladed)’’ and ‘‘Straight-bladed’’ VAWTs. Sketches of these twovariations of Darrieus concepts are shown in Figs. 2 and 3, respectively. The Darrieus-typeVAWTs are basically lift force driven wind turbines. The turbine consists of two or moreaerofoil-shaped blades which are attached to a rotating vertical shaft. The wind blowingover the aerofoil contours of the blade creates aerodynamic lift and actually pulls theblades along. The troposkien shape eggbeater-type Darrieus VAWT, which minimizes thebending stress in the blades, were commercially deployed in California in the past.

Fig. 2. Curved-blade (or ‘‘Egg-beater’’ type) Darrieus VAWT.

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Fig. 3. Straight-bladed Darrieus VAWT.


In the small-scale wind turbine market, the simple straight-bladed Darrieus VAWT,often called giromill or cyclo-turbine, is more attractive for its simple blade design. Thisconfiguration fall into two categories: fixed pitch and variable pitch. It has been found out


from the previous research activities that fixed pitch VAWTs provide inadequate startingtorque [6]. Contemporary variable pitch blade configuration has potential to overcome thestarting torque problem but it is overly complicated, rendering it impractical for smallercapacity applications. Majority of the previously conducted research activities on VAWTfocused on straight bladed VAWTs equipped with symmetric airfoils (like NACA 4-digitseries of 0012, 0015, 0018) which were unable to self-start. This inability to self-start is dueto several factors (like technical, inadequate research work & funding), and the mostdominant ones are due to aerodynamic factors. According to Internet survey, there arehandfuls of commercial straight-bladed VAWTs products, but no reliable informationcould be obtained from an independent source regarding the performance of theseproducts and the claims made by the manufacturers are yet to be authentically verified.At present, development of large-scale straight-bladed VAWT is limited to the research

stage only, although large eggbeater Darrieus wind turbine had reached the marketcommercially in the past before disappearing away later. However, in the small-scale windturbine market, the simple straight-bladed Darrieus seems to be more cost effective thanthe eggbeater Darrieus as few companies had marketed this type of wind turbine before,i.e. the Pinson/Asi cycloturbine which utilized an end tail for changing pitch. Thisparticular giromill model was stated in Drees’ [8] research paper of having 3.6m diameterand 2.4m height. With 3 blades at chord length of 29 cm, the rotor has solidity of 0.24.Another pitch changing research prototype was built by Grylls et al. [9]. It has a diameterof 2.4m and a height of 1.6m. Using 3 blades with a chord length of 14.5 cm only, therotor has a solidity of 0.18. Wind tunnel results for this prototype indicated the rotor wasable to self-start at wind speed of 3.5m/s, provided the pitch angle change is larger thanplus or minus 41.

3.3. H-Rotors

H-Rotors, as shown in Fig. 4, were developed in the UK through the research carriedout during the 1970–1980s when it was established that the elaborated mechanisms used tofeather the straight-bladed Darrieus VAWT blades were unnecessary. It was found outthat the drag/stall effect created by a blade leaving the wind flow would limit the speed thatthe opposing blade (in the wind flow) could propel the whole blade configuration forward.The H-Rotor was therefore self-regulating in all wind speeds reaching its optimalrotational speed shortly after its cut-in wind speed.

4. General mathematical expressions for aerodynamic analysis of straight-bladed Darrieus-

type VAWTs

Though the straight-bladed Darrieus-type VAWT is the simplest type of wind turbine,its aerodynamic analysis is quite complex. Before comparative analysis of the mainaerodynamic models, the general mathematical expressions, which are common to most ofthe aerodynamic models, are described in this section.

4.1. Variation of local angle of attack

The flow velocities in the upstream and downstream sides of the Darrieus-type VAWTsare not constant as seen in Fig. 5. From this figure one can observe that the flow is

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Fig. 4. H-Rotor-type VAWT.


considered to occur in the axial direction. The chordal velocity component Vc and thenormal velocity component Vn are, respectively, obtained from the following expressions:

V c ¼ Roþ V a cos y, (1)

Vn ¼ V a sin y, (2)

where Va is the axial flow velocity (i.e. induced velocity) through the rotor, o is therotational velocity, R is the radius of the turbine, and y is the azimuth angle. Referring toFig. 5, the angle of attack (a) can be expressed as

a ¼ tan�1Vn

V c

� �. (3)

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Fig. 5. Flow velocities of straight-bladed Darrieus-type VAWT.


Substituting the values of Vn and Vc and non-dimensionalizing,

a ¼ tan�1sin y

ðRo=V1Þ=ðV a=V1Þ þ cos y

� �, (4)

where V1 is the freestream wind velocity. If we consider blade pitching then,

a ¼ tan�1sin y

ðRo=V1Þ=ðV a=V1Þ þ cos y

� �� g, (5)

where g is the blade pitch angle.

4.2. Variation of local relative flow velocity

The relative flow velocity (W) can be obtained as (Fig. 5),

W ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2

c þ V 2n

q. (6)

Inserting the values of Vc and Vn (Eqs. (1) and (2)) in Eq. (6), and non-dimensionalizing,one can find velocity ratio as,

W

V1¼

W

Va:

V a

V1¼

Va

V1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRoV1

�V au

V1

� �þ cos y

� �2þ sin2y

s. (7)

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Fig. 6. Force diagram of a blade airfoil.


4.3. Variation of tangential and normal forces

The directions of the lift and drag forces and their normal and tangential componentsare shown in Fig. 6. The tangential force coefficient (Ct) is basically the difference betweenthe tangential components of lift and drag forces. Similarly, the normal force coefficient(Cn) is the difference between the normal components of lift and drag forces. Theexpressions of Ct and Cn can be written as

Ct ¼ C1 sin a� Cd cos a, (8)

Cn ¼ C1 cos aþ Cd sin a. (9)

The net tangential and normal forces can be defined as

F t ¼ Ct12rCHW2, (10)

Fn ¼ Cn12rCHW2, (11)

where r is the air density, C is the blade chord and H is the height of the turbine.

4.4. Calculation of total torque

Since, the tangential and normal forces represented by Eqs. (10) and (11) are for anyazimuthal position, so, they are considered as a function of azimuth angle y. Averagetangential force (F ta) on one blade can be expressed as

F ta ¼1

2p

Z 2p

0

F tðyÞdy. (12)

The total torque (Q) for the number of blades (N) is obtained as

Q ¼ NF taR. (13)


4.5. Power output

The total power (P) can be obtained as

P ¼ Q � o. (14)

5. Computational models for Darrieus-type straight-bladed VAWT

In the past, several mathematical models, based on several theories, were prescribedfor the performance prediction and design of Darrieus-type VAWTs by differentresearchers. The key components of all the computational models can be broadlydescribed as:

�
calculations of local relative velocities and angle of attacks at different tip speed ratiosand azimuthal (orbital) positions; � calculation of ratio of induced to freestream velocity (Va=V1) considering the blade/
blade-wake interaction;
� mathematical expressions based on different approaches (Momentum, Vortex or
Cascade principles) to calculate normal and tangential forces;
� ‘Pre-Stall airfoil characteristics’ (Cl, Cd & Cm) for the attached regime at different
Reynolds numbers;
� ‘Post-Stall Model’ for Stall Development and Fully Stalled regimes; � ‘Finite Aspect Ratio consideration’; � ‘Dynamic Stall Model’ to account for the unsteady effects; � ‘Flow Curvature Model’ to consider the circular blade motion.
According to literature survey, the most studied and best validated models can bebroadly classified into three categories—(1) Momentum model, (2) Vortex model and (3)Cascade model. It should be noted that not all the models consider all the key componentsdescribed above. Descriptions of these three main categories of VAWT computationalmodels are presented below.

5.1. Momentum model

Different momentum models (also specified as Blade Element/Momentum or BEMmodel) are basically based on calculation of flow velocity through turbine by equating thestreamwise aerodynamic force on the blades with the rate of change of momentum of air,which is equal to the overall change in velocity times the mass flow rate. The force is alsoequal to the average pressure difference across the rotor. Bernoulli’s equation is applied ineach streamtube. The main drawback of these models is that they become invalid for largetip speed ratios and also for high rotor solidities because the momentum equations in theseparticular cases are inadequate [10]. Over the years, several approaches were attempted toutilize this concept, which are discussed briefly in the following headings.

5.1.1. Single streamtube model

In 1974 Templin proposed the single streamtube model which is the first and most simpleprediction method for the calculation of the performance characteristics of a Darrieus-type


VAWTs [11]. In this model the entire turbine is assumed to be enclosed within a singlestreamtube as shown in Fig. 7. This model first incorporated the concept of the windmillactuator disc theory into the analytical prediction model of a Darrieus-type VAWT. In thistheory the induced velocity (rotor axial flow velocity) is assumed to be constantthroughout the disc and is obtained by equating the streamwise drag with the change inaxial momentum.

In the assumption, the actuator disc is considered as the surface of the imaginary bodyof revolution. It is assumed that the flow velocity is constant throughout the upstream anddownstream side of the swept volume. This theory takes into account the effect of airfoilstalling on the performance characteristics. The effects of geometric variables such as bladesolidity and rotor height–diameter ratio have been included in the analysis. The effect ofzero-lift-drag coefficient on the performance characteristics has also been included. Windshear effect cannot be incorporated into the model.

Now, according to Gluert Actuator Disk theory, the expression of the uniform velocitythrough the rotor is

Va ¼V1 þ Vw

2, (15)

where Vw is the wake velocity. All the calculations in this model are performed for a singleblade whose chord equals the sum of the chords of the actual rotor’s blades. The

Fig. 7. Schematic of single streamtube model.


streamwise drag force (FD) due to the rate of change of momentum is

FD ¼ _m � V1 � Vwð Þ, (16)

where _m (¼ ArV a) is the mass flow rate. The rotor drag coefficient (CDD) is defined as

CDD ¼FD

1=2rAV 2a

. (17)

From Eqs. (16) and (17), we can find that

CDD ¼ 4V1 � Va

V a

� �(18)

and

Va

V1¼

1

1þ CDD=4

� �. (19)

The overall torque and power coefficient of the VAWT can be determined from Eqs.(13) and (14) by utilizing the expression of Va=V1 derived in Eq. (19) above.This model can predict the overall performance of a lightly loaded wind turbine but

according to the inquest, it always predicts higher power than the experimental results. Itdoes not predicts the wind velocity variations across the rotor. These variations graduallyincrease with the increase of the blade solidity and tip speed ratio. In 1980, Noll and Hampresented an analytical method for the performance prediction of a cyclically pitchedstraight-bladed vertical-axis wind turbine using the single streamtube model [12]. Theyadded the effect of strut drag, turbulent wake state and dynamic stall to their analyticalmethod.

5.1.2. Multiple streamtube model

In 1974, Wilson and Lissaman [13] introduced the Multiple streamtube model which wasan improvement to single streamtube model. In this model the swept volume of the turbineis divided into a series of adjacent, aerodynamically independent parallel streamtubes asshown in Fig. 8. The blade element and momentum theories are then employed for eachstreamtube. In their model they considered the flow as inviscid and incompressible for thecalculation of the induced velocity through the streamtube. As a result, there appears onlythe lift force in the calculation of the induced velocity. Wilson and Lissaman [13]considered the theoretical lift for their calculation, which is given by

Cl ¼ 2p sin a. (20)

In this model, the induced velocity ratio can be found from the following expression:

Va

V1¼ 1�

k

2:Nc

R:R$

V1: sin y

� �, (21)

where k is a factor found through iteration. In this model, the induced velocity varies overthe frontal disc area both in the vertical and horizontal directions [14]. Atmospheric windshear can be included in this model. However, this model still is inadequate in itsdescription of flow field and it can be applied only for a fast running lightly loaded windturbine.

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Fig. 8. Schematic of multiple streamtube model.


In 1975, Strickland [15] presented another multiple streamtube model for a Darrieus-type VAWT. In this model, induced velocity is found by equating the blade elementalforces (including airfoil drag) and the change in the momentum along each streamtube.The wind shear effects have also been included in the calculation of the model. This modelpredicts the overall performance reasonably, especially when the rotor is lightly loaded. Itdisplays improvement over the single streamtube model.

The basic difference between Wilson’s and Strickland’s models is that Wilson used thetheoretical lift force only in the calculation of induced velocity while Strickland added theeffect of drag force as well for the similar calculation. Among these two models, Wilson’smodel gives fast convergence while Strickland’s model gives slow convergence due toadded complicacy.

Another theory based on the multiple streamtube model including the effects of airfoilgeometry, support struts, blade aspect ratio, turbine solidity and blade interference waspresented by Muraca et al. [16]. The effect of flow curvature is evaluated by considering theflow over a flat plate. They derived an expression of lift distribution on the plate with thevariable angle of attack from the leading to the trailing edge points of the flat plate andaveraged the distributed lift force over the whole surface of the flat plate. According tothem, the effect of flow curvature on the performance characteristics is insignificant for alow chord to radius ratio.

In 1977, Sharpe gave an elaborated description of the multiple streamtube model in areport. The principal idea of his model is similar to Strickland’s model. Additionally, heincorporated the effect of Reynolds number into the calculation [17]. In 1980, anotherimproved version of the multiple streamtube model was presented by Read and Sharpe [18]


for VAWT. In their model the parallel streamtube concept is dispensed with and theexpansion of the streamtube is included. It is strictly applicable to low solidity lightlyloaded wind turbines with large aspect ratio (H/c). It can predict the instantaneousaerodynamic blade forces and the induced velocities better than those by the conventionalmultiple streamtube model. But the prediction of overall power coefficients cannot bemade with reasonable accuracy. It usually gives lower power than that obtainedexperimentally.

5.1.3. Double-multiple streamtube model

In 1981, Paraschivoiu [19] introduced double multiple streamtube theory for theperformance prediction of a Darrieus wind turbine. As shown in Fig. 9, in this model thecalculation is done separately for the upstream and downstream half cycles of the turbine.At each level of the rotor, the upstream- and downstream-induced velocities are obtainedusing the principle of two actuator discs in tandem. The concept of the two actuator discsin tandem for a Darrieus wind turbine was originally given by Lapin [20]. For both theupstream and downstream half cycles vertical variation of the induced velocity (like that inthe multiple stream tube model) is considered while in the horizontal direction inducedvelocity is assumed to be constant (like that of a single streamtube model). For theupstream half-cycle, the wake velocity is represented by

V e ¼ V1i 2Vau

V1i� 1

� �¼ V1i 2uu � 1ð Þ, (22)

where V1i is the local ambient wind velocity (which is different at different heights of theturbine bladed due to the effect of wind shear), Vau is the induced velocity anduu( ¼ V au=V1i) is the interference factor for the upstream half-cycle. For the downstreamhalf-cycle of the rotor, V e is the input velocity. The induced velocity for the downstreamhalf-cycle is V ad which can be written as

V ad ¼ udV e ¼ ud 2uu � 1ð ÞV1i, (23)

Fig. 9. Schematic of double-multiple streamtube model.


where, ud( ¼ V ad=V e) is the interference factor for the downstream half-cycle. Thestreamtube induced velocity is calculated by a double iteration, one for each part of therotor.

The double-multiple streamtube model with constant and variable interference factors(induced velocity ratios), including secondary effects for a Darrieus wind rotor wasexamined by Paraschivoiu et al. [21]. They found a relatively significant influence of thesecondary effects, namely, the blade geometry and profile type, the rotating tower and thepresence of struts and aerodynamically spoilers, especially at high tip speed ratios. Theyconsidered the variation of the induced velocity as a function of azimuth angle that gives amore accurate calculation of the aerodynamic loads. In the paper presented byParaschivoiu and Delclaux [22], they made improvements in the double-multiplestreamtube model. They considered the induced velocity variation as a function of theazimuth angle for each streamtube.

The double-multiple streamtube model gives better correlation between the calculatedand experimental results, especially for the local aerodynamic blade forces with themultiple streamtube models. However, this model gives over prediction of power for a highsolidity turbine and there appears to be a convergence problem for the same type ofturbine especially in the downstream side and at the higher tip speed ratio.

5.2. Vortex model

The Vortex models are basically potential flow models based on the calculation of thevelocity field about the turbine through the influence of vorticity in the wake of the blades.The turbine blades are represented by bound or lifting-line vortices whose strengths aredetermined using airfoil coefficient datasets and calculated relative flow velocity and angleof attack.

A simple representation of the vortex system associated with a blade element is shown inFig. 10. The VAWT blade element is replace by a ‘‘bound’’ vortex filament sometimescalled ‘‘substitution’’ vortex filament or a lifting line. The strengths of the boundvortex and each trailing tip vortex are equal as a consequence of the Helmholtz theoremsof voticity [23]. According to Fig. 10, the strengths of the shed vortex systems havechanged on several occasions. On each of these occasions, a spanwise vortex is shed whose

Fig. 10. Vortex system for a single blade element.


strength is equal to the change in the bound vortex strength as dictated by Kelvin’stheorem [23].The fluid velocity at any point in the flow field is the sum of the undisturbed wind stream

velocity and the velocity induced by the entire vortex filaments in the flow filed. Thevelocity induced at a point in the flow field by a single vortex filament can be obtained fromthe Biot–Savart law, which relates the induced velocity to the filament strength. Referringto the case shown in Fig. 11, for a straight vortex filament of strength G and length l,induced velocity ~Vp at a point P on the filament is given by,

~Vp ¼ ~eG4pd

cos y1 � cos y2ð Þ, (24)

where d is the minimum distance of the point P from the vortex filament, ~e and ~r are theunit vectors. It should be noted that if point P should happen to lie on the vortex filament,Eq. (24) yields indeterminate results, since ~e cannot be defined and the magnitude of ~Vp isinfinite. The velocity induced by a straight filament on itself is, in fact, equal to zero.In order to allow closure of the Vortex model, a relationship between the bound vortex

strength and the velocity induced at a blade segment must be obtained. A relationshipbetween the lift L per unit span on a blade segment and the bound vortex strength GB isgiven by the Kutta–Joukowski law. The lift can also be formulated in terms of the airfoillift coefficient CL. Equating these two expressions for the lift, yields the requiredrelationship between the bound vortex strength and the induced velocity at a particularblade segment as,

GB ¼12cCLW , (25)

where, c is the blade chord. After determining the induced velocity distribution, it becomesstraight-forward to obtain performance characteristics of VAWT as described in Section 4.

Fig. 11. Velocity induced at a point by a vortex filament.


In 1975, Larsen [24] first introduced the idea of Vortex model. The Vortex system for asingle blade element of a VAWT. He used his Vortex model for the performance predictionof a cyclogiro windmill. His model is a two-dimensional one but if the vortex trailing fromthe rotor blade tips is considered, it may be said that it is not strictly two-dimensional.However, in his model angle of attack is assumed small, as a result, the stall effect isneglected.

Fanucci and Walters [25] presented a two-dimensional Vortex model applicable to astraight-bladed VAWT. In their analysis they considered the angle of attack very smallwhich eliminates the stall effect. Holme [26] presented a Vortex model for a fast runningvertical-axis wind turbine having a large number of straight, very narrow blades and a highheight–diameter ratio (in order to make a two-dimensional flow assumption). The analysisis valid for a lightly loaded wind turbine only. Wilson [27] also introduced a two-dimensional vortex analysis to predict the performance of a giromill. In his method he didnot take the stall effect into account, because the angle of attack was assumed to be small.

In 1979, Strickland et al. [28] presented an extension of the Vortex model which is athree-dimensional one and the aerodynamic stall is incorporated into the model. Theypresented the experimental results for a series of two-dimensional rotor configurations.Their calculated values show more or less good correlation with the experimental resultsfor the instantaneous blade forces and the near wake flow behind the rotor. Strickland etal. [29] made improvements on the prior Vortex model (quasi-steady Vortex model). Thelatest model is termed as the dynamic Vortex model, since, in this model the dynamiceffects are included. The improvements over the prior model are that it includes thedynamic stall effect, pitching circulation and added mass effect. They repeated theexperiment with the test model as is mentioned in Ref. [30] and found appreciablevariations with the prior results. The correlation with their calculated values by thedynamic Vortex model and the latest experimental results of the local blade forces andwake velocities seem to be reasonable in some cases.

In 1984, Cardona [31] incorporated the effect of flow curvature following the methodgiven by Migliore et al. [32] into the original aerodynamic Vortex model presented byStrickland et al. [30]. They also added a modified form of the dynamic stall effects. Theyfound an improved correlation with the calculated and experimental results for theinstantaneous aerodynamic blade forces and the overall power coefficients.

The main disadvantage of Vortex model is that it takes too much computation time.Furthermore, this model still rely on significant simplifications, like potential flow isassumed in the wake and the effect of viscosity in the blade aerodynamics is includedthrough empirical force coefficients [33].

5.3. Cascade model

The periodic equidistant arrangement of several blades or vanes of turbomachinaries iscalled a cascade. Hence, the cascade is the basic element of the turbomachine, and cascadeflow is the essential physical phenomenon for the operation of the machine [34]. TheCascade model was proposed by Hirsch and Mandal [35] to apply the cascade principles,widely used for turbomachinaries, for the analysis of VAWTs for the first time. In thismodel, the blade airfoils of a turbine are assumed to be positioned in a plane surface(termed as the cascade) with the blade interspace equal to the turbine circumferentialdistance divided by the number of blades as shown in Fig. 12. The relationship between the

ARTICLE IN PRESS

Fig. 12. Development of blade into a cascade configuration.


wake velocity and the free stream velocity is established by using Bernoulli’s equationwhile the induced velocity is related to the wake velocity through a particular semi-empirical expression.In this model, the aerodynamic characteristics of each element of the blade are obtained

independently, like the double-multiple streamtube theory, for the upwind and downwindhalves of the rotor considering the local Reynolds number and the local angle of attack asshown in Fig. 13. After determination of the local relative flow velocity and the angle ofattack, the VAWT is developed into a cascade configuration that is shown in Fig. 12. Thecascade is considered in a plane normal to the turbine axis. If the blade represented by A atan azimuth angle y is considered as the reference blade, the flow conditions on the othertwo blades represented by B and C, are assumed to be equal to those of the referenceblade. This process is continued for one complete revolution of the reference blade with astep of dy.To find the induced velocity, a relationship between the wake velocity and the induced

velocity is introduced. For the upstream side this is expressed as

V au

V1¼

V e

V1

� �ki

(26)

and for the downstream side, this is expressed as

V ad

V e¼

Vw

V e

� �ki

, (27)

where V e and Vw are the wake velocities in the upstream and downstream side. The valueof the exponent ki is found from a fit of experimental results. The induced velocity ratio for

ARTICLE IN PRESS

Fig. 13. Horizontal section of a straight-bladed Darrieus-type VAWT with flow velocities in the upstream and

downstream sides.


the downstream side can be written in the form,

Vad

V1¼

V ad

V e

V e

V1¼

V e

V1

Vw

V e

� �ki

. (28)

The expression of the exponent ki is written in accordance with Ref. [35] as

ki ¼ ð0:425þ 0:332sÞ, (29)

where s ¼ NC=R . The final expression for the overall torque is found from,

Q ¼ rR3 H

R

Z 2p

0

ðW 22 sin a2 cos a2 �W 2

1 sin a1 cos a1Þdy, (30)

where W 1 and W 2 are the relative velocities in the cascade inlet and outlet. Detaildescription of this model can be found from Hirsh and Mandal [35].

The Cascade model can predict the overall values of both low and high solidity turbinesquite well. It takes reasonable computation time. It does not make any convergenceproblem even at the high tip speed ratios and high solidities. The instantaneous bladeforces calculated by this model show improved correlation in comparison to thosecalculated by the conventional Momentum model. The theory also incorporated theeffect of the local Reynolds number variation at different azimuth angles (orbital position),


zero-lift-drag coefficients, finite aspect ratios and the flow curvature effect in thecalculation process.Subsequently, to improve the analytical capability of this model, two important effects

of the dynamic stall and flow curvature with blade pitching were taken into account byMandal and Burton [36]. The calculated values of the wake velocities after thesemodifications become comparable with those by the complex dynamic Vortex model.

6. Conclusions

Several aerodynamic models have been analyzed in this paper which are applied forbetter performance prediction and design analysis of straight-bladed Darrieus-typeVAWT. At present the most widely used models are the double-multiple streamtubemodel, free-Vortex model and the Cascade model. It has been found that, each of thesethree models has their strengths and weaknesses. Though among these three models, theVortex models are considered to be the most accurate models according to severalresearchers, but they are computationally very expensive and in some cases they sufferfrom convergence problem. It has also been found that the double-multiple streamtubemodel is not suitable for high tip speed ratios and high-solidity VAWT. On the other hand,the Cascade model gives smooth convergence even in high tip speed ratios and high solidityVAWT with quite reasonable accuracy.

Acknowledgments

This work was financially supported by Natural Sciences and Engineering ResearchCouncil of Canada (NSERC). Authors would also like to thank Umm Tahzin for drawingthe sketches for this paper.

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